Quantum dynamics on tight binding small world networks and flat band spectra

We investigate spectral features of quantum walks on graphs using tight binding hamiltonians that range from nearest neighbor coupling to fully connected models. For equal strength bonds in the few bond limit we find a transition from square root DOS to a semicircular Wigner-like one while close to the mean field limit a fully degenerate band appears. When long range bonds are weaker than nearest neighbor ones we have a similar transition that however re-enters to the square root spectrum at very high long range bond fillings. These findings may have some significance in complex lattices with flat -like bands.